Abstract
In this paper we establish the definition of the generalized inverse A_{T,S}^{(2)} which is a \{2\} inverse of a matrix A with prescribed image T and kernel S over an associative ring, and give necessary and sufficient conditions for the existence of the generalized inverse A_{T,S}^{(1,2)} and some explicit expressions for A_{T,S}^{(1,2)} of a matrix A over an associative ring, which reduce to the group inverse or \{1\} inverses. In addition, we show that for an arbitrary matrix A over an associative ring, the Drazin inverse A_{d}, the group inverse A_{g} and the Moore–Penrose inverse A^{\dagger }, if they exist, are all the generalized inverse A_{T,S}^{(2)}.
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2000 Mathematics Subject Classification:
primary 15A33, 15A09

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